Path properties of volatility: A statistical analysis

我们发展统计理论来推断随机过程波动的路径特性。波动率是描述随机过程演化中不确定性的关键量。它的路径性质决定了波动的光滑性和相关性结构，从而决定了最优估计方法、预测技术和波动的持续性。尽管波动率的路径特性对于当前实证工作的应用和主题特别感兴趣，但迄今为止还没有统计基础。最近，有重要的贡献，先进的估计和测试程序的路径性质的随机过程的直接观察。关键的区别和困难在于波动性是潜在的，因此不能直接观察到。基于涉及预先估计的波动性的统计数据，这两个申请人在第一篇文章中开始了这一新方向的工作，重点是变点分析。利用最具创新性的贡献，从直接观察的波动率估计和路径属性的推断，我们的目标是建立一个新的研究链与最佳的统计方法。目前，在文献中提出了相互矛盾的波动模型。为了建立适当的波动率模型，需要更多的知识波动的路径特性。我们考虑了丰富的一类随机波动过程，以调和不同的程式化事实。在这个项目中开发的理论将提供哪些模型是合适的，而且，如果路径属性是持久的或随时间变化的证据。对这项工作特别感兴趣的是金融计量经济学。波动率是描述价格演变中的市场风险的流行概念。因此，可靠的波动性估计是风险分析的关键要素。我们主要关注的是在最高可用记录频率下的日内高频数据。建模和分析这种高频数据变得越来越重要，因为目前近70%的交易量归因于高频交易。与此同时，需要考虑到具体的市场摩擦，从而导致嘈杂的价格观察。我们设计了两种噪声规范的方法：经典的中心市场微观结构噪声模型和不规则的噪音适合于记录的价格从限价订单簿。这是在三个相辅相成的一揽子工作中进行的。一个工作包，提供新的方法基础，是由申请人。在理想化的框架中，我们确定识别某些路径属性所需的最小信息量，并构建有效的恢复技术。这两个博士项目专注于两个相关但不同的噪声观测模型的优化方法。更复杂的结构，这些现实的模型要求几个创新，以解决识别的路径特性的波动。

We develop statistical theory to infer path properties of the volatility of a stochastic process. Volatility is the key quantity to describe uncertainty in the evolution of a stochastic process. Its path properties determine the smoothness and dependence structure of volatility and thus optimal estimation methods, forecasting techniques and the persistence of volatility. Even though path properties of volatility are of particular interest for applications and subject of current empirical work, there is so far no statistical groundwork. Recently, there have been important contributions to advance estimation and testing procedures on path properties of stochastic processes from direct observations. The key difference and difficulty is that volatility is latent and thus not directly observable. Based on statistics involving pre-estimated volatility, the two applicants started to work in this new direction in a first article focusing on change-point analysis. Exploiting the most innovative contributions regarding volatility estimation and inference on path properties from direct observations, we aim to establish a novel strand of research with optimal statistical approaches. Currently, conflicting models for volatility are put forward in the literature. In order to build adequate volatility models, more knowledge on the path properties of volatility is required. We consider a rich class of random volatility processes to reconcile different stylized facts. The theory developed in this project will provide evidence about which models are suitable and, moreover, if path properties are persistent or time-varying. Particular interest in this work is motivated by financial econometrics. Volatility is the prevailing concept to describe market risk in price evolutions. Reliable volatility estimates are thus key ingredients for risk analysis. Our main focus is on intra-day high-frequency data at highest available recording frequencies. Modeling and analysing such high-frequency data becomes more and more important as much volume, currently almost 70 %, is attributed to high-frequency trading. At the same time, specific market frictions need to be taken into account, inducing noisy price observations. We design approaches for two noise specifications: The classical centred market microstructure noise model and irregular noise suitable for recorded prices from limit order books. This is pursued in three complementary work packages. One work package, providing novel methodological groundwork, is addressed by the applicants. In an idealized framework, we determine the minimal amount of information necessary to identify certain path properties and construct efficient techniques for their recovery. The two doctoral projects focus on optimal methods for the two related yet different noisy observation models. The more involved structure of these realistic models ask for several innovations to address the identification of the path properties of volatility.